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For a positive integer $n$ not divisible by $211$ , let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$ . Find the remainder when $$ \sum_{n=1}^{210} nf(n) $$ is divided by $211$ .
*Proposed by ApraTrip* | 48 | 4/8 |
Rhombus $PQRS^{}_{}$ is inscribed in rectangle $ABCD^{}_{}$ so that vertices $P^{}_{}$, $Q^{}_{}$, $R^{}_{}$, and $S^{}_{}$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB^{}_{}=15$, $BQ^{}_{}=20$, $PR^{}_{}=30$, and $QS^{}_{}=40$. Let $\frac{m}{n}$, in lowest terms, denote the perimeter of $ABCD^{}_{}$. Find $m+n^{}_{}$. | 677 | 7/8 |
Determine the lateral sides of an isosceles trapezoid if its bases and area are equal to 8 cm, 14 cm, and 44 cm², respectively. | 5\, | 1/8 |
For an integer \( n \geq 3 \), we say that \( A = \left( a_1, a_2, \ldots, a_n \right) \) is an \( n \)-list if every \( a_k \) is an integer in the range \( 1 \leq a_k \leq n \). For each \( k = 1, \ldots, n-1 \), let \( M_k \) be the minimal possible non-zero value of \( \left| \frac{a_1 + \ldots + a_{k+1}}{k+1} - \frac{a_1 + \ldots + a_k}{k} \right| \), across all \( n \)-lists. We say that an \( n \)-list \( A \) is ideal if
\[ \left| \frac{a_1 + \ldots + a_{k+1}}{k+1} - \frac{a_1 + \ldots + a_k}{k} \right| = M_k \]
for each \( k = 1, \ldots, n-1 \).
Find the number of ideal \( n \)-lists. | 4(n-1) | 1/8 |
In a castle, there are 16 identical square rooms arranged in a $4 \times 4$ square. Sixteen people—some knights and some liars—each occupy one room. Liars always lie, and knights always tell the truth. Each of these 16 people said, "At least one of my neighboring rooms houses a liar." Rooms are considered neighbors if they share a common wall. What is the maximum number of knights that could be among these 16 people? | 12 | 2/8 |
Find the integers \( n \) greater than or equal to 2 such that, if \( a \) denotes the smallest prime divisor of \( n \), one can find a positive divisor of \( n \) denoted \( d \) such that \( n = a^3 + d^3 \). | 16,72,520 | 1/8 |
Black and white balls are arranged in a circle, with black balls being twice as many as white balls. It is known that among the pairs of neighboring balls, there are three times as many pairs of the same color as there are pairs of different colors. What is the minimum number of balls that could have been arranged? | 24 | 5/8 |
Given that the plane vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ satisfy $|\boldsymbol{\alpha} + 2\boldsymbol{\beta}| = 3$ and $|2\boldsymbol{\alpha} + 3\boldsymbol{\beta}| = 4$, find the minimum value of $\boldsymbol{\alpha} \cdot \boldsymbol{\beta}$. | -170 | 4/8 |
The side length of the larger base of a right quadrangular truncated pyramid is $a$. The lateral edge and the pyramid's diagonal form angles $\alpha$ and $\beta$ with the plane of the base, respectively. Find the area of the smaller base of the pyramid. | \frac{^2\sin^2(\alpha-\beta)}{\sin^2(\alpha+\beta)} | 1/8 |
Let
\[ f(x) = \ln x - \frac{1}{2}ax^2 - 2x \quad \text{for } a \in [-1, 0), \]
and suppose \( f(x) < b \) holds in the interval \((0, 1]\). Determine the range of the real number \( b \). | (-\frac{3}{2},+\infty) | 1/8 |
A pebble is shaped as the intersection of a cube of side length 1 with the solid sphere tangent to all of the cube's edges. What is the surface area of this pebble? | \frac{6 \sqrt{2}-5}{2} \pi | 1/8 |
Given the vertices of a pyramid with a square base, and two vertices connected by an edge are called adjacent vertices, with the rule that adjacent vertices cannot be colored the same color, and there are 4 colors to choose from, calculate the total number of different coloring methods. | 72 | 7/8 |
In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts is the winner; finally, the giraffe that wins in the majority of districts is declared the winner of the final. The Tall giraffe won. What is the minimum number of voters who could have voted for the Tall giraffe? | 30 | 4/8 |
In space, $n$ planes are drawn. Each plane intersects with exactly 1999 others. Find all possible values of $n$. | 3998 | 7/8 |
Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\]Find $|f(0)|$. | 72 | 3/8 |
If the set \( S = \{1, 2, 3, \cdots, 16\} \) is arbitrarily divided into \( n \) subsets, there must exist some subset that contains elements \( a, b, \) and \( c \) (which can be the same) such that \( a + b = c \). Find the maximum value of \( n \).
**Note**: If the subsets \( A_1, A_2, \cdots, A_n \) of set \( S \) satisfy the following conditions:
1. \( A_i \neq \varnothing \) for \( i = 1, 2, \cdots, n \);
2. \( A_i \cap A_j = \varnothing \);
3. \( \bigcup_{i=1}^{n} A_i = S \),
then \( A_1, A_2, \cdots, A_n \) are called a partition of set \( S \). | 3 | 4/8 |
Given the circle $C: x^2 + y^2 - (6 - 2m)x - 4my + 5m^2 - 6m = 0$, and a fixed line $l$ passing through the point $A(1, 0)$, for any real number $m$, the chord intercepted by circle $C$ on line $l$ always has a constant length $A$. Find the constant value of $A$. | \frac{2\sqrt{145}}{5} | 7/8 |
The students of a school went on a trip, for which two buses were hired. When the buses arrived, 57 students got on the first bus and only 31 on the second. How many students need to move from the first bus to the second so that the same number of students are transported in both buses?
(a) 8
(b) 13
(c) 16
(d) 26
(e) 31 | 13 | 1/8 |
Find all positive integers $m$ for which there exist three positive integers $a,b,c$ such that $abcm=1+a^2+b^2+c^2$ . | 4 | 1/8 |
A company receives apple and grape juices in identical standard containers and produces a cocktail (mixture) of these juices in identical standard cans. Last year, one container of apple juice was sufficient to make 6 cans of cocktail, and one container of grape juice was sufficient to make 10 cans of cocktail. This year, the ratio of juices in the cocktail (mixture) has changed, and now one standard container of apple juice is sufficient to make 5 cans of cocktail. How many cans of cocktail can now be made from one standard container of grape juice? | 15 | 7/8 |
Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$ . Suppose that $BD$ and $AC$ intersect at a point $E$ . The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$ , so that $C$ lies between $E$ and $F$ . Given that $AE=6$ , $EC=4$ , $BE=2$ , and $BF=12$ , find $DA$ . | 2\sqrt{42} | 4/8 |
Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat? | 27 | 3/8 |
For $a>0$ , denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$ .
Find the maximum area of $S(a)$ . | \frac{8\sqrt{2}}{3} | 7/8 |
Given that $0 < α < \frac {π}{2}$, and $\cos (2π-α)-\sin (π-α)=- \frac { \sqrt {5}}{5}$.
(1) Find the value of $\sin α+\cos α$
(2) Find the value of $\frac {2\sin α\cos α-\sin ( \frac {π}{2}+α)+1}{1-\cot ( \frac {3π}{2}-α)}$. | \frac {\sqrt {5}-9}{5} | 7/8 |
Lei Lei bought some goats and sheep. If she had bought 2 more goats, the average price of each sheep would increase by 60 yuan. If she had bought 2 fewer goats, the average price of each sheep would decrease by 90 yuan. Lei Lei bought $\qquad$ sheep in total. | 10 | 1/8 |
A regular quadrilateral pyramid is intersected by a plane passing through a vertex of the base perpendicular to the opposite lateral edge. The area of the resulting cross-section is half the area of the pyramid's base. Find the ratio of the height of the pyramid to the lateral edge. | \frac{1+\sqrt{33}}{8} | 1/8 |
The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$ , while its charge is $\frac12$ at pH $9.6$ . Charge increases linearly with pH. What is the isoelectric point of glycine? | 5.97 | 7/8 |
Given that point $A(1,1)$ is a point on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $F\_1$, $F\_2$ are the two foci of the ellipse such that $|AF\_1|+|AF\_2|=4$.
(I) Find the standard equation of the ellipse;
(II) Find the equation of the tangent line to the ellipse that passes through $A(1,1)$;
(III) Let points $C$ and $D$ be two points on the ellipse such that the slopes of lines $AC$ and $AD$ are complementary. Determine whether the slope of line $CD$ is a constant value. If it is, find the value; if not, explain the reason. | \frac{1}{3} | 1/8 |
Barry has three sisters. The average age of the three sisters is 27. The average age of Barry and his three sisters is 28. What is Barry's age?
(A) 1
(B) 30
(C) 4
(D) 29
(E) 31 | 31 | 1/8 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = \sqrt{2}$, $|\overrightarrow{b}| = 2$, and $(\overrightarrow{a} - \overrightarrow{b}) \perp \overrightarrow{a}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} | 7/8 |
Let \( f_0(x) = |x| \), \( f_1(x) = \left| f_0(x) - 1 \right| \), \( f_2(x) = \left| f_1(x) - 2 \right| \), \ldots, \( f_n(x) = \left| f_{n-1}(x) - n \right| \). The function \( f_n(x) \) has exactly two common points with the x-axis. What is the area enclosed by this function curve and the x-axis? | \frac{4n^3+6n^2-1+(-1)^n}{8} | 1/8 |
Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$? | 7 | 2/8 |
An employer hired a worker for a year and promised to give him 12 rubles and a coat. However, the worker wanted to leave after 7 months. Upon settlement, he received the coat and 5 rubles. How much did the coat cost? | 4.8 | 7/8 |
There are \( n \) vectors in space such that any pair of them forms an obtuse angle. What is the maximum possible value of \( n \)? | 4 | 4/8 |
Let \(a, b, c\) be arbitrary real numbers such that \(a > b > c\) and \((a - b)(b - c)(c - a) = -16\). Find the minimum value of \(\frac{1}{a - b} + \frac{1}{b - c} - \frac{1}{c - a}\). | \frac{5}{4} | 6/8 |
Let $\clubsuit$ and $\heartsuit$ be whole numbers such that $\clubsuit \times \heartsuit = 48$ and $\clubsuit$ is even, find the largest possible value of $\clubsuit + \heartsuit$. | 26 | 2/8 |
In a football tournament, each team is supposed to play one match against each of the other teams. However, during the tournament, half of the teams were disqualified and did not participate further. As a result, a total of 77 matches were played, and the disqualified teams managed to play all their matches against each other, with each disqualified team having played the same number of matches. How many teams were there at the beginning of the tournament? | 14 | 1/8 |
Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, and outcomes of tosses being independent, calculate the probabilities of winning Game A and Game B. | \frac{1}{4} | 1/8 |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.
$(1)$ Find $p$;
$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | 20\sqrt{5} | 4/8 |
If we pick uniformly a random square of area 1 with sides parallel to the $x$ and $y$ axes that lies entirely within the 5-by-5 square bounded by the lines $x=0$, $x=5$, $y=0$, and $y=5$ (the corners of the square need not have integer coordinates), what is the probability that the point $(x, y)=(4.5, 0.5)$ lies within the square of area 1? | \frac{1}{64} | 6/8 |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and $\sin A$, $\sin B$, and $\sin C$ form a geometric sequence. When $B$ takes the maximum value, the maximum value of $\sin A + \sin C$ is _____. | \sqrt{3} | 3/8 |
After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $7$ team members?
$\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$ | \textbf{(B)}11 | 1/8 |
The numbers \( a \) and \( b \) satisfy the condition \( a + b > 0 \). Which of the following inequalities are necessarily true?
a) \( a^{5} b^{2} + a^{4} b^{3} \geqslant 0 \)
b) \( a^{4} b^{3} + a^{3} b^{4} \geqslant 0 \)
c) \( a^{21} + b^{21} > 0 \)
d) \( (a+2)(b+2) > ab \)
e) \( (a-3)(b-3) < ab \)
f) \( (a+2)(b+3) > ab + 5 \) | d | 7/8 |
Allen and Brian are playing a game in which they roll a 6-sided die until one of them wins. Allen wins if two consecutive rolls are equal and at most 3. Brian wins if two consecutive rolls add up to 7 and the latter is at most 3. What is the probability that Allen wins? | \frac{5}{12} | 2/8 |
A set $M$ consists of $n$ elements. Find the greatest $k$ for which there is a collection of $k$ subsets of $M$ such that for any subsets $A_{1},...,A_{j}$ from the collection, there is an element belonging to an odd number of them | n | 7/8 |
Given points P(-2, -2), Q(0, -1), and a point R(2, m) is chosen such that PR + PQ is minimized. What is the value of the real number $m$? | -2 | 7/8 |
Let \( P \) be a point inside regular pentagon \( ABCDE \) such that \( \angle PAB = 48^\circ \) and \( \angle PDC = 42^\circ \). Find \( \angle BPC \), in degrees. | 84 | 1/8 |
$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc. | 504 | 3/8 |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[ f(x)f(y) = f(xy) + 2023 \left( \frac{1}{x} + \frac{1}{y} + 2022 \right) \]
for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s$. | \frac{4047}{2} | 5/8 |
Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed? | 13 | 7/8 |
The factors of $x^4+64$ are: | (x^2-4x+8)(x^2+4x+8) | 2/8 |
A finite sequence of numbers \( x_{1}, x_{2}, \ldots, x_{N} \) possesses the following property:
\[ x_{n+2} = x_{n} - \frac{1}{x_{n+1}} \text{ for all } 1 \leq n \leq N-2. \]
Find the maximum possible number of terms in this sequence if \( x_{1} = 20 \) and \( x_{2} = 16 \). | 322 | 2/8 |
In a single-elimination tournament consisting of $2^{9}=512$ teams, there is a strict ordering on the skill levels of the teams, but Joy does not know that ordering. The teams are randomly put into a bracket and they play out the tournament, with the better team always beating the worse team. Joy is then given the results of all 511 matches and must create a list of teams such that she can guarantee that the third-best team is on the list. What is the minimum possible length of Joy's list? | 45 | 1/8 |
Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull? | \frac{\sqrt{3}}{2} | 2/8 |
In a scalene triangle \(ABC\), one of the angles is equal to the difference between the other two angles, and one of the angles is twice another angle. The angle bisectors of \(\angle A\), \(\angle B\), and \(\angle C\) intersect the circumcircle of the triangle at points \(L\), \(O\), and \(M\) respectively. Find the area of triangle \(LOM\) if the area of triangle \(ABC\) is equal to 8. If the answer is not an integer, round it to the nearest integer. | 11 | 1/8 |
In the arithmetic sequence $\left\{ a_n \right\}$, $a_{15}+a_{16}+a_{17}=-45$, $a_{9}=-36$, and $S_n$ is the sum of the first $n$ terms.
(1) Find the minimum value of $S_n$ and the corresponding value of $n$;
(2) Calculate $T_n = \left| a_1 \right| + \left| a_2 \right| + \ldots + \left| a_n \right|$. | -630 | 3/8 |
A point is randomly thrown onto the segment $[11, 18]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+2k-99\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | \frac{2}{3} | 3/8 |
Given a right quadrilateral pyramid \( V-ABCD \) with the height equal to half the length of \( AB \), \( M \) is the midpoint of the lateral edge \( VB \), and \( N \) is a point on the lateral edge \( VD \) such that \( DN=2VN \). Find the cosine of the angle between the skew lines \( AM \) and \( BN \). | \frac{\sqrt{11}}{11} | 5/8 |
Compute the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{1+\operatorname{tg} x \cdot \cos 2 x}{1+\operatorname{tg} x \cdot \cos 5 x}\right)^{\frac{1}{x^{3}}}
$$ | e^{\frac{21}{2}} | 2/8 |
Convert the binary number $1110011_2$ to its decimal equivalent. | 115 | 7/8 |
Let \( g(x) \) be the function defined on \(-2 \le x \le 2\) by the formula
\[ g(x) = 2 - \sqrt{4 - x^2}. \]
If a graph of \( x = g(y) \) is overlaid on the graph of \( y = g(x) \), then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth? | 2.28 | 6/8 |
In triangle \(ABC\), point \(K\) is taken on side \(AC\) such that \(AK = 1\) and \(KC = 3\), and point \(L\) is taken on side \(AB\) such that \(AL:LB = 2:3\). Let \(Q\) be the intersection point of lines \(BK\) and \(CL\). The area of triangle \(AQC\) is 1. Find the height of triangle \(ABC\), dropped from vertex \(B\). | 1.5 | 1/8 |
Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She starts with a triangle $A_{0} A_{1} A_{2}$ where angle $A_{0}$ is $90^{\circ}$, angle $A_{1}$ is $60^{\circ}$, and $A_{0} A_{1}$ is 1. She then extends the pasture. First, she extends $A_{2} A_{0}$ to $A_{3}$ such that $A_{3} A_{0}=\frac{1}{2} A_{2} A_{0}$ and the new pasture is triangle $A_{1} A_{2} A_{3}$. Next, she extends $A_{3} A_{1}$ to $A_{4}$ such that $A_{4} A_{1}=\frac{1}{6} A_{3} A_{1}$. She continues, each time extending $A_{n} A_{n-2}$ to $A_{n+1}$ such that $A_{n+1} A_{n-2}=\frac{1}{2^{n}-2} A_{n} A_{n-2}$. What is the smallest $K$ such that her pasture never exceeds an area of $K$? | \sqrt{3} | 1/8 |
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\]
Find $\max_{a\leq 1983} M(a).$ | 121 | 2/8 |
(Convert the following binary number to decimal: 101111011<sub>(2)</sub>) | 379 | 6/8 |
Ship \( P \) spots ship \( Q \), which is moving in a direction perpendicular to \( PQ \), maintaining its course. Ship \( P \) chases \( Q \), always heading directly towards \( Q \); the speed of both ships at any moment is the same (but can change over time). It is clear without calculations that \( P \) travels along a curved path; if the chase lasts long enough, the trajectory of the pursuing ship and the trajectory of the fleeing ship eventually become nearly identical. What will the distance \( PQ \) be then, if initially it was 10 nautical miles? | 5 | 1/8 |
Let \( P \) and \( A \) denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of \(\frac{P^{2}}{A}\). | 45 | 4/8 |
On a \(6 \times 6\) chessboard, we randomly place counters on three different squares. What is the probability that no two counters are in the same row or column? | 40/119 | 7/8 |
In an isosceles trapezoid \(ABCD\) (\(BC \parallel AD\)), the angles \(ABD\) and \(DBC\) are \(135^{\circ}\) and \(15^{\circ}\) respectively, and \(BD = \sqrt{6}\). Find the perimeter of the trapezoid. | 9 - \sqrt{3} | 4/8 |
Hooligan Vasya is unhappy with his average math grade, which dropped below 3. As a measure to sharply raise his grade, he accessed the school journal and changed all his failing grades to C's. Prove that even after this, his average grade will not exceed 4. | 4 | 7/8 |
On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The k-th building has exactly k (k=1, 2, 3, 4, 5) workers from Factory A, and the distance between two adjacent buildings is 50 meters. Factory A plans to build a station on this street. To minimize the total distance all workers from Factory A have to walk to the station, the station should be built at a distance of meters from Building 1. | 150 | 4/8 |
Let \( S = \{1, 2, \cdots, n\} \). \( A \) is an arithmetic sequence with a positive common difference and at least two terms, all of which are in \( S \). After adding any other element from \( S \) to \( A \), it should not form a new arithmetic sequence with the same common difference as \( A \). Find the number of such sequences \( A \). (In this context, a sequence with only two terms is also considered an arithmetic sequence). | \lfloor\frac{n^2}{4}\rfloor | 1/8 |
Two cars start from the same location at the same time, moving in the same direction at a constant speed. Each car can carry a maximum of 24 barrels of gasoline, and each barrel of gasoline allows a car to travel 60km. Both cars must return to the starting point, but they do not have to return at the same time. The cars can lend gasoline to each other. To maximize the distance one car can travel away from the starting point, the other car should turn back at a distance of $\boxed{360}$ km from the starting point. | 360 | 3/8 |
Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( A M : A A_{1} = 1 : 2, B N : B B_{1} = 1 : 3, \) and \( C K : C C_{1} = 1 : 4 \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( M N K P \) if the volume of the prism is 16. | 4 | 6/8 |
Given the set $A=\{m+2, 2m^2+m\}$, if $3 \in A$, then the value of $m$ is \_\_\_\_\_\_. | -\frac{3}{2} | 1/8 |
The chord $AB$ of a circle with radius 1 and center at $O$ is the diameter of a semicircle $ACB$ located outside the first circle. It is clear that the point $C$ of this semicircle, which protrudes the furthest, lies on the radius $ODC$ perpendicular to $AB$. Determine $AB$ so that the segment $OC$ has the maximum length. | \sqrt{2} | 7/8 |
Does there exist a positive integer $m$ such that the equation $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{a b c} = \frac{m}{a + b + c}$ has infinitely many solutions in positive integers? | 12 | 1/8 |
The ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Under the threat of severe punishment, he decreed that each family should have no more than one girl. As a result, in this country, each woman's last - and only last - child was a girl because no woman dared to have more children after giving birth to a girl. What proportion of boys comprised the total number of children in this country, assuming the chances of giving birth to a boy or a girl are equal? | 2/3 | 1/8 |
Consider the function \( f: \mathbb{N} \rightarrow \mathbb{Z} \) satisfying, for all \( n \in \mathbb{N} \),
(a) \( |f(n)| = n \)
(b) \( 0 \leq \sum_{k=1}^{n} f(k) < 2n \).
Evaluate \( \sum_{n=1}^{2018} f(n) \). | 2649 | 1/8 |
Arrange the positive integers whose digits sum to 4 in ascending order. Which position does the number 2020 occupy in this sequence? | 28 | 7/8 |
A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is $810 \mathrm{~cm}^{2}$. | 36 | 1/8 |
Pyotr has a deck of 36 cards (4 suits with 9 cards in each). He chooses half of the cards (whichever he wants) and gives them to Vasya, keeping the other half for himself. In each subsequent turn, players alternately place one card face-up on the table, starting with Pyotr. If Vasya can respond to Pyotr's card by placing a card of the same suit or rank, Vasya earns 1 point. What is the maximum number of points Vasya can be guaranteed to earn? | 15 | 1/8 |
Let the function \( f(x) = x^2 - x + 1 \). Define \( f^{(n)}(x) \) as follows:
$$
f^{(1)}(x) = f(x), \quad f^{(n)}(x) = f\left(f^{(n-1)}(x)\right).
$$
Let \( r_{n} \) be the arithmetic mean of all the roots of \( f^{(n)}(x) = 0 \). Find \( r_{2015} \). | \frac{1}{2} | 7/8 |
Given \( f_{1}(x)=|1-2 x| \) for \( x \in [0,1] \) and \( f_{n}(x)=f_{1}(f_{n-1}(x)) \), determine the number of solutions to the equation \( f_{2005}(x)=\frac{1}{2} x \). | 2^{2005} | 7/8 |
$A$ and $B$ ran around a circular path with constant speeds. They started from the same place and at the same time in opposite directions. After their first meeting, $B$ took 1 minute to go back to the starting place. If $A$ and $B$ need 6 minutes and $c$ minutes respectively to complete one round of the path, find the value of $c$. | 3 | 7/8 |
Let point \( O \) be a point inside triangle \( ABC \) that satisfies the equation
\[
\overrightarrow{OA} + 2 \overrightarrow{OB} + 3 \overrightarrow{OC} = 3 \overrightarrow{AB} + 2 \overrightarrow{BC} + \overrightarrow{CA}.
\]
Then, find the value of \(\frac{S_{\triangle AOB} + 2 S_{\triangle BOC} + 3 S_{\triangle COA}}{S_{\triangle ABC}}\). | \frac{11}{6} | 7/8 |
The Fibonacci sequence is defined as follows: \( F_{0}=0, F_{1}=1 \), and \( F_{n}=F_{n-1}+F_{n-2} \) for all integers \( n \geq 2 \). Find the smallest positive integer \( m \) such that \( F_{m} \equiv 0 \pmod{127} \) and \( F_{m+1} \equiv 1 \pmod{127} \). | 256 | 2/8 |
Xiao Ming places some chess pieces into a $3 \times 3$ grid. Each small square within the grid can have zero, one, or more chess pieces. After counting the number of chess pieces in each row and each column, he obtains 6 different sums. What is the minimum number of chess pieces needed? | 8 | 6/8 |
Four spheres touch each other at different points, and their centers lie in a plane Π. Another sphere S touches all these spheres. Prove that the ratio of its radius to the distance from its center to plane Π is equal to \(1: \sqrt{3}\). | \frac{1}{\sqrt{3}} | 1/8 |
Select 3 numbers from the range 1 to 300 such that their sum is exactly divisible by 3. How many such combinations are possible? | 1485100 | 5/8 |
Ms. Garcia weighed the packages in three different pairings and obtained weights of 162, 164, and 168 pounds. Find the total weight of all four packages. | 247 | 1/8 |
Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters exist? Points for the problem: 8. | 86 | 3/8 |
In the trapezoid \(ABCD\), the base \(AB\) is three times longer than the base \(CD\). On the base \(CD\), point \(M\) is taken such that \(MC = 2MD\). \(N\) is the intersection point of lines \(BM\) and \(AC\). Find the ratio of the area of triangle \(MNC\) to the area of the entire trapezoid. | \frac{1}{33} | 7/8 |
Given $2$ red and $2$ white balls, a total of $4$ balls are randomly arranged in a row. The probability that balls of the same color are adjacent to each other is $\_\_\_\_\_\_$. | \frac{1}{3} | 6/8 |
The probability that Class A will be assigned exactly 2 of the 8 awards, with each of the 4 classes (A, B, C, and D) receiving at least 1 award is $\qquad$ . | \frac{2}{7} | 1/8 |
The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where $a$, $b$, and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$.
| 447 | 7/8 |
A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
$\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$ | \textbf{(E)} | 1/8 |
Find the number of integer pairs \((x, y)\) that satisfy the system of inequalities:
\[
\left\{\begin{array}{l}
2x \geq 3y \\
3x \geq 4y \\
5x - 7y \leq 20
\end{array}\right.
\] | 231 | 2/8 |
A telephone number has the form $\text{ABC-DEF-GHIJ}$, where each letter represents
a different digit. The digits in each part of the number are in decreasing
order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore,
$D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd
digits; and $A + B + C = 9$. Find $A$.
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$ | \textbf{(E)}\8 | 1/8 |
Given the polynomial \( f(x) = a_{2007} x^{2007} + a_{2006} x^{2006} + \cdots + a_{3} x^{3} + 2x^2 + x + 1 \), prove that \( f(x) \) has at least one complex root. | 1 | 6/8 |
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